Binary Addition using 2’s Complement

When negative numbers are expressed in binary addition using 2’s
complement the addition of binary numbers becomes easier. This operation is
almost similar to that in 1’s complement system and is explained with examples
given below:

A.
Addition of a positive number and a negative number.

We consider the following cases.

Case I: When the positive
number has a greater magnitude

In this case the carry which will be generated is discarded and the
final result is the result of addition.

The
following examples will illustrate this method in binary addition using 2’s complement:

In a 5-bit register find the sum
of the following by using 2’s complement:

(i) -1011 and -0101

Solution:

+ 1 0 1 1
⇒ 0 1 0 1 1

- 0 1 0 1
⇒ 1 1 0 1 1 (2’s complement)

(Carry 1 discarded) 0 0 1 1 0

Hence the sum
is + 0110.

(ii) + 0111 and – 0011.

Solution:

+ 0 1 1 1
⇒ 0 0 1 1 1

- 0 0 1 1
⇒ 1 1 1 0 1

(Carry 1 discarded) 0 0 1 0 0

Hence the sum is + 0100.

Case II: When the negative
number is greater.

When the negative numbers is greater no carry will be generated in the
sign bit. The result of addition will be negative and the final result is
obtained by taking 2’s complement of the magnitude bits of the result.

The
following examples will illustrate this method in binary addition using 2’s complement:

In a 5-bit register
find the sum of the following by using 2’s complement:

(i) + 0 0 1 1 and - 0
1 0 1

Solution:

+ 0 0 1 1
⇒ 0 0 0 1 1

- 0 1 0 1
⇒ 1 1 0 1 1 (2’s complement)

1 1 1 1 0

2’s complement
of 1110 is (0001 + 0001) or 0010.

Hence the
required sum is - 0010.

(ii) + 0 1 0 0 and - 0 1 1 1

Solution:

+ 0 1 0 0
⇒ 0 0 1 0 0

- 0 1 1 1
⇒ 1 1 0 0 1 (2’s complement)

1 1 1 0 1

2’s complement
of 1101 is 0011.

Hence the required sum is – 0011.

B. When the numbers are negative.

When two
negative numbers are added a carry will be generated from the sign bit which
will be discarded. 2’s complement of the magnitude bits of the operation will
be the final sum.

The
following examples will illustrate this method in binary addition using 2’s complement:

In a 5-bit
register find the sum of the following by using 2’s complement: